\(N_{max}\)  is a LL cutoff as introduced in [1]}, and has been used previous to our work on Lnadau mWSMs [2]}. In fact, as pointed out in [3]}, the correct choice of limits decides whether we are in the static limit or uniform limit of the CME. The expression was originally derived in the context of gluons in [1]} by defining an appropriate thermodynamic potential. While it is a useful exercise to understand this derivation, we focus our attention towards evaluating this object for our model. Since the expression inside the integral is a total derivative the answer is obtained as
\(\alpha = -\frac{J_z}{B} &= \frac{e^2}{4\pi ^2 \hbar ^2} \left[ n E_0^{-} + \sum _{m=n}^{N_{max}} \sum _{\gamma = \pm } E_m^{-,\gamma } \right]_{\Lambda }^{\Lambda }.\) 
